Chapter 3

Copyright © 2009 Pearson Prentice Hall. All rights reserved.

Chapter 3Stock and Bond Valuation: Annuities and Perpetuities

Copyright © 2009 Pearson Prentice Hall. All rights reserved.3-2

Chapter 3 Outline

3.1 Perpetuities3.2 Annuities3.3 The Four Formulas SummarizedAppendix: Advanced Material3.4 Projects With Different Lives and Rental

Equivalents3.5 Perpetuity and Annuity Derivations


Stock and Bond ValuationAnnuities and Perpetuities

• The present value formula is the primary way to find value.

• Formulas can provide a shortcut to having to write out all the cash flows to infinity. Whew!

• The perpetuity formula values a series of infinite payments.

• The annuity formula values payments that span a time period.

• Both formulas need the payments or cash flows to be equal dollar amounts every period.


Stock and Bond Valuation:Example: $2 Perpetuity at a 10% Interest Rate

• What would you have to invest today to receive the same value as $2 of interest paid to you each year forever, starting next year, if the constant interest rate is 10%?

• You could add $2 X 1/(1+r)t for every t until infinity on a spreadsheet but that might take a while….probably more time than you have.

• Using the perpetuity formula, we can calculate the answer:

• You find the answer is $20.

• And you don’t miss meals……just keep r and C constant; the formula works.

PV C1

r

PV $2

.10PV $20


Stock and Bond ValuationPerpetuities

• A perpetuity is a project with a set of constant cash flows that repeats forever.

• If the cost of capital (r) and the cash flow per period is the same from today until infinity, then you need the perpetuity formula.

• Perpetuity Formula:

• Since the first cash flow occurs next year, C1 is the proper cash flow.

• The formula is the mathematical upper limit of a infinite series.

PV C1

r


Stock and Bond ValuationSimple Perpetuity Table for $2 at a rate of 10%

TABLE 3.1 Perpetuity Stream of $2 with Interest Rate r = 10%


Stock and Bond ValuationGrowing Perpetuity Formula

• What if the cash flow grows by the same percentage every year?

• Now we need a growing perpetuity formula (the Gordon Growth Model).

• Growing perpetuity or Gordon Growth Model:

• We are investing to receive the firm’s future cash flows, but if the cash flows grow at a constant rate, we reduce the discount rate in the denominator by the growth rate.

• Growth is a benefit to us. • By reducing the denominator, ‘g’ will increase the PV of the cash flows.

• This formula is very useful when valuing stocks since corporations have infinite lives and many pay steadily increasing dividends.

• …..the average investor hopes dividends increase!

PV C1

r g


Stock and Bond ValuationGrowing Perpetuity Example: $2 growing at 5%, r = 10%

• What if next year’s $2 payment grew at 5% every year forever when investors require a return of 10%?

• We could grow cash flow from $2 to $2.10 to $2.205 to forever, and then discount…..but we might miss a meal, so use the formula.

• The growing perpetuity formula:

• We find the answer is $40, double the “no growth” answer. • Note: growth is very important to valuation!

• We assume r and g are constant to infinity. If g isn’t greater or equal to the discount rate, we can use this formula on stocks with dividends.

PV C1

r g

PV $2

.10 .05PV $40


Stock and Bond ValuationGrowing Perpetuity Example: $2 growing at 5%, r = 10%

TABLE 3.2 Perpetuity Stream with C1 = $2, Growth Rate g = 5%, and Interest Rate r = 10%


Stock and Bond ValuationBusiness Valuation with Gordon Growth Model

• What is today’s value of a stable business that grows with the inflation rate of 2% and is expected to have $1,000,000 in profit next year when the firm’s cost of capital (required return) is 8%?

• Use the growing perpetuity formula as follows:

• We find today’s value of the business to be $16,666,667.

• Since many good and bad things can happen to the business, consider this one estimate of its value. Its true value could be higher or lower if more factors were considered.

PV C1

r g

PV $1,000,000

8% 2% $1,666,667

BusinessValue $1,666,667


Stock and Bond ValuationStock Valuation Example with a Growing Perpetuity

• Applying the Gordon growth model, we can find today’s value of a stock with a $10 dividend next year that’s growing by 5% (forever) when we expect a return equal to the S&P500 long-run return of 10%.

• Using the growing perpetuity formula, we can calculate the stock’s value:

• We find the value to be $200 per share.

• Rearranging the Gordon growth formula, we can observe that:

• We can find the value of the firm’s cost of capital r if we know a stock’s dividend, its stock’s price, and its growth rate.

PV C1

r g

PV $10

10% 5%$200

Stock Value $200

r C1

PV g r

Div1

P g


Stock and Bond ValuationFinding Cost of Capital with the Gordon Growth Model

• Looking at GE’s stock data for October of 2004, we observe:

• Dividend yield ( Div1 / Stock price )

• Growth of dividends

• Using the Gordon Growth Formula,

• Cost of capital, or r

• We find that investors in GE must be using a cost of capital of 12%.

r C1

PV g

r Div1

P g

r 2.4% 9.6%

r 12%

2.4%

9.6%

12%


Stock and Bond ValuationThe Gordon Growth Model and Forward Earnings Yield, 1/P/E

• For the GE data, if we assume the earnings yield equals the dividend yield, we find:

• Forward P/E= 18.5 (forward looks at next year’s EPS)• Growth in earnings = 6.3%• 1 / Forward P/E = E1 / P and we’ll substitute it for Div1 / Stock Price

• Next, we rearrange the Gordon Growth Model to find cost of capital, r:

• Thus, investors in GE who use earnings as the source of their valuation (and dividends), find a cost of capital of 11.7%, which is close to the 12% we found before.

• Warning! Use of these simple relationships will do your portfolio harm! The world of investing is not so simple. Industries and markets change too much to allow investors to assume the relationships stay fixed until infinity, or even next month.

r C1

PV g

r E1

P g

r 1

18.5 6.3% 5.4% 6.3%

r 11.7%


Stock and Bond ValuationAnnuities

• Annuities value a series of equal dollar cash flows per period over time.

• Rather than separately calculating the present value of $5 in Year 1, Year 2, and Year 3 at 10%, we can do it all in one formula.

• The Annuity Formula:

• This formula works so you don’t miss any meals when someone asks how much that mortgage payment for 360 months is worth.

PV C1

r 1

1

(1 r)T

PV $5

.10 1

1

(1 .10)3

Solving,

PV $12.43


Stock and Bond ValuationAnnuity Application: 30-Year, Fixed-Rate Mortgage

• Most mortgages on homes are repaid over 360 months (30 years) in equal payments and have a fixed interest rate (think annuity formula). Take the annual loan rate and divide it by 12 to find the monthly rate used over the 360 months.

• What is your monthly payment if you borrow $500,000 and the annual rate is 7.5%?

• We are solving for the payment C1 for a 360-month loan (use PV annuity formula).

• Find monthly r first, which is .075/12 = .00625.

• Your monthly payment equals $3,496.07 and that includes the loan’s principal and interest. Most payments have some real estate taxes and home insurance, too. Some even have you pay insurance to protect the lender; that’s PMI, private mortgage insurance. If you only borrow 80% of the value, then you don’t pay PMI.

PV C1

r 1

1

(1 r)T

$500,000 C1

.0625 1

1

(1 .0625)360

Solving,

C1 $3,496.07


Stock and Bond ValuationBonds and Annuities Example: Bond Valuation

• A typical American coupon bond pays semiannual interest (also called coupon payments) until its maturity date when the bond’s principal or face value is repaid.

• A typical 2-year semiannual-payment bond’s cash flows are:

6 months 1 year 1.5 years 2 yearscoupon coupon coupon coupon + face value

• The coupon yield describes the annual interest payments. It is calculated on the value of the principal (face value). A bond with 3% coupon yield (cy) and a principal value of $100,000 has coupon payments every year of $3,000.

• To find the semiannual payments, divide by 2 to find payments of $1,500 every 6 months.

• Bond coupon payment = ½ X Cy X face = ½ X 3% X $100,000

• Bond coupon payment = $1,500


Stock and Bond Valuation Bonds Example: Bond Valuation -- continued

• The 2-year ( 4-period ) bond cash flows look like this on a timeline:

Period 1 Period 2 Period 3 Period 46 months 1 year 1.5 years 2 years$1500 $1500 $1500 $1500 + $100,000

• To value these cash flows you would need to know the bond’s discount rate, called a yield-to-maturity, and convert it to a semiannual rate.

• Assume the Yield to Maturity is 5.0%; this is the bond’s discount rate.

• We’ll find our semiannual discount rate r using

• The semiannual rate is 2.47%.

Semi-annual Rate 1YTM 1 r

Semi-annual Rate 1.05 1 .0247 r

1YTM 1 r


Stock and Bond Valuation:Bonds Example: Bond Valuation -- continued 2

• We have a 2-year (4-period) bond with the following cash flows:

PeriodCash flow x Discount rate= PVs1 1,500 1/1.02471

2 1,500 1/1.02472

3 1,500 1/1.02473

4 101,500 1/1.02474

• Find the PV of the cash flows, sum, and you’ll find the value of the bond:

• Value equals $96,348.25.

• Since the bond only promises to pay 3% and the discount rate is 5% annually, the bond sells at a discount to its principal or face value.

• Remember that the coupon rate (cy) is the company’s promised rate, and the yield to maturity (YTM) is the market’s desired or discount rate.


Stock and Bond ValuationLevel-Coupon Bond, 5 Years to Maturity, 3% Coupon, 5% YTM

Step 1: Write down the bond’s payment stream.

Step 2: Find the appropriate cost of capital for each payment.


Stock and Bond ValuationLevel-Coupon Bond, 5 Years to Maturity, 3% Coupon, 5% YTM

Step 3: Compute the discount factor—it is 1/(1+r1).


Stock and Bond ValuationFormula Method: Bond 5-Year Maturity, 3% Coupon, 5% YTM

• What if the 3% level-coupon bond matures in 5 years and has a YTM of 5.0%? We could do a table or we could use the annuity formula as a shortcut.

• Our bond is a ten-payment annuity ($1500 every 6 months), and the $100,000 face value is paid with the final coupon. The semiannual discount rate is 2.47%.

• Value of bond = PV of 10-period coupon annuity + PV of face value from 10 periods

The value of the bond equals $91,497.32.

• The bond offers less return than desired by investors, so they pay less than face value.

Value of bond Coupon1

r 1

1

(1 r)T

Face

(1 r)T

Value of bond

1

2 cy Face

r 1

1

(1 r)T

Face

(1 r)T

Value of bond

1

2 3% $1000

.0247 1

1

(1 .0247)10

$1000

(1 .0247)10

Value of bond $1500

.0247 1

1

(1 .0247)10

$1000

(1 .0247)10

Value of bond $91, 497.32


Stock and Bond ValuationAn Aside on Discount, Premium, and Par Bonds

• A discount bond has a coupon rate that is less than its yield-to-maturity. If a bond offers less than what investors want, they will pay less for it.

• Discount Bond (Coupon rate < YTM)$91,000 valuation and $100,000 face amount

• A premium bond has a coupon rate greater than its YTM. Some bonds pay more than the market requires. Those bonds sell at a premium to face.

• Premium Bond (Coupon rate > YTM)$105,000 valuation and $100,000 face amount

• A par bond has a coupon rate equal to its yield-to-maturity. The bond sells for its face (or principal) value.

• Par Bond (Coupon rate = YTM)$100,000 valuation and $100,000 face amount


Stock and Bond ValuationThe Four Formulas (three we use regularly)

• The Perpetuity Formula:

• The Gordon Growth Model or Growing Perpetuity Formula:

• The Simple Annuity Formula:

• The fourth formula (from pension finance) is the Growing Annuity Formula:

• These four formulas are useful for many different types of corporate decisions.

PV C1

r 1 1

(1 r)T

PV C1

(r g) 1 (1g)T

(1 r)T

PV C1

(r g)

PV C1

r


Stock and Bond Valuation:The Four Payoff Streams and Their Present Values

FIGURE 3.1 The Four Payoff Streams and Their Present Values

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